Théorie des groupes/Group Theory Théorie des

Théorie des groupes/Group Theory
Théorie des Nombres/Number Theory
A FORMAL THEORY OF EISENSTEIN SERIES
Kâzım İlhan İkeda
May 1995
Abstract. The definition of the Eisenstein series E (Z, s; k, Γ) on the symplectic and unitary groups G is well-known. We introduce a series E(Z, λ) whose construction involves a
5-tuple {G, P, M, W, ²} satisfying certain conditions. We prove that, if λ : Mn×2n (k) −→ C
is a locally-constant function, then the series E(Z, λ) is a finite linear combination of Gtransforms of the Eisenstein series E (Z, s; k, Γ) whose coefficients are products of certain
Hecke L-functions.
UNE THÉORIE FORMELLE DES SÉRIES d’EISENSTEIN
Résumé. Sur les groupes symplectique et unitaire G la définition des séries d’Eisenstein
E (Z, s; k, Γ) est bien connue. Dans cette note nous introduisons une série E(Z, λ) dont la
construction fait intervenir un 5-tuplet {G, P, M, W, ²} satisfaisant à certaines conditions.
Nous démontrons que, si λ : Mn×2n (k) −→ C est une fonction localement constante, alors
la série E(Z, λ) est une combinaison linéaire finie des G-transformés des séries d’Eisenstein
E (Z, s; k, Γ) dont les coefficients sont produits de certaines fonctions L de Hecke.
Version française abrégée. §1. Considérons la donnée (que nous appelons une donnée
d’Eisenstein dans la suite) définie par un 5-tuplet {G, P, M, W, ²} où ² ∈ W est un
ensemble non-vide, G et M sont des groupes de transformations sur W , et P est un
sous-groupe de G satisfaisant aux conditions suivantes:
(i) G opère transitivement à droite sur W et M opére fidèlement à gauche de W . Ces
actions sur W sont compatibles comme dans (1); (ii) T² ⊂ P où T² désigne le groupe
d’isotropie de ² dans G; (iii) Il existe un morphisme surjectif d : P −→ M tel que le carré
(2) est commutatif, où r² est défini par l’action à droite de P sur ² et l² est l’application
définie par l’action à gauche de M sur ²; (iv) Il existe un G-espace H et un groupe
G = GL∗ (C) avec un facteur d’automorphie j : G × H −→ G à valeurs dans G tel que la
condition (3) est satisfaite pour tous π ∈ P et z ∈ H; où δ : M −→ G est un morphisme.
§2. (Tous les calculs du §2 sont formels.) Soit Γ un sous-groupe de G et soit ∆ un sousgroupe de M vérifiant la condition (4). Etant donné un caractère ψ : Γ → T, vérifiant
(7), où B désigne un système complet de représentants de P \G/Γ, on considère les séries
E(z, λ, ψ) comme dans (8), où λ : W → C est une application quelconque vérifiant (6).
La série d’Eisenstein E (z, Γ, ψ) de G (relative à Γ et à ψ) est définie comme dans (9).
On démontre au §2 (Théorème 2.3) que la série E(z, λ, ψ) est une combinaison linéaire
des β-transformés des séries d’Eisenstein E (z, βΓβ −1 , ψβ ) pour β ∈ B.
This report announces some of the results of the author’s Ph.D. thesis, Princeton University 1993.
The author heartily thanks Goro Shimura for his help and encouragement all through author’s graduate
study. He also thanks the referee for his valuable suggestions.
Typeset by AMS-TEX
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§3. Dans cette section, F désigne un corps de nombres totalement réel de degré m
et K désigne un CM -corps ayant F pour sous-corps totalement réel maximal. Pour
uniformiser notre exposé, le symbole k désigne soit F , soit K s’il n’y a aucune confusion et Ok désigne l’anneau des entiers de k. On note a l’ensemble a F des premiers
archimédiens de F ou bien d’un CM -type fixé de K, et h désigne l’ensemble des premiers
non-archimédiens de F . Pour X ∈ Mn×sµ
(K), nous posons
X ∗ = t X ι , où ι : K −→ K
¶
0 −1n
est l’involution de Galois sur F , et Jn =
∈ M2n (Q). Nous considérons les
1n
0
groupes algébriques Sp(n, F ) et SU (n, K) définis sur F que nous noterons simplement
∼
par Gk . Nous introduisons Gk comme dans (12). Notons Pk le sous-groupe parabolique
de Gk correspondant à la partition (n, n), Qk la composante de Levi et Rk le radi∼
∼
∼
∼
cal unipotent de Pk . Les sous-groupes P k , Qk et Rk de Gk sont définis de la même
manière. Soit Wk = {α ∈ Mn×2n (k) | αJn α∗ = 0, rang(α) = n}. On démontre que
∼
∼
{Gk , P k , GLn (k), Wk , (0n , 1n )} est une donnée d’Eisenstein. Pour un sous-groupe congru∼
ence Γ de Gk (cf [6] ppQ424), définissons la série d’Eisenstein E (Z, s; k, Γ) comme dans
(15), en supposant que ν∈aa det(µν (xν , zν ))k = 1 pour tout x ∈ Pk ∩Γ pour que la somme
ait un sens. Soit ∆ un sous-groupe de congruence de GLn (k) suffisamment petit pour
que δ(∆) = 1. Notons par S(V ) l’espace des fonctions localement constantes définies sur
un espace vectoriel V sur Q à valeurs dans C. Considérons un élément λ ∈ S(M n×2n (k)),
tel que λ(∆wΓ) = λ(w) pour w ∈ Mn×2n (k). Suivant la notation de (5), introduisons la
somme E(Z, λ) comme dans (16).
∼
Théorème
3.1. Soit Γ un sous-groupe de congruence de Gk satisfaisant à la condition
Q
k
a
a ∈ H et soit ∆
a det(µν (xν , zν )) = 1 pour tout x ∈ Pk ∩ Γ et tout Z = (zν )ν∈a
ν∈a
un sous-groupe de congruence de GLn (k) tel que δ(∆) = 1. Soit λ : Mn×2n (k) −→ C
une fonction localement constante avec la propriété λ(∆wΓ) = λ(w) pour w ∈ M n×2n (k).
Alors, la série E(Z, λ) est une combinaison linéaire finie des β-transformés E (Z, s :
k, βΓβ −1 ) de la série d’Eisenstein dont les coefficients sont des sommes de produits de
séries L de Hecke.
1.PRELIMINARIES. Consider the data (which we call an Eisenstein datum in the
sequel) defined by a 5-tuple {G, P, M, W, ²}, where ² ∈ W is a (non-empty) set, G and M
transformation groups on W and P a subgroup of G satisfying the following conditions:
(i) G acts transitively on the right of W and M acts faithfully on the left of W . These
actions on W are compatible in the following sense:
(1)
(mw)g = m(wg), for every m ∈ M , w ∈ W and g ∈ G.
(ii) T² ⊂ P , where T² denote the isotropy group of ² in G.
(iii) There exist a surjective morphism d : P −→ M such that the square
r
(2)
P −−−²−→


idP y
d
W
x
l
²
P −−−−→ M
2
is commutative, where r² : P −→ W is defined via the right action of P on ² and
l² : M −→ W is the map defined by the left action of M on ².
(iv) There exists a G-space H and a group G = GL∗ (C) with a G-valued factor of automorphy j : G × H −→ G (that is the cocycle relation j(g1 g2 , z) = j(g1 , g2 (z))j(g2 , z), for
g1 , g2 ∈ G and for z ∈ H is satisfied) such that
(3)
j(π, z) = δ(d(π)),
for every π ∈ P and z ∈ H; where δ : M −→ G is a morphism.
Let {G, P, M, W, ²} be an Eisenstein datum. Define the mapping τ : G → W by
g 7→ τ (g) = ²g, for g ∈ G. Then we readily have the following
Lemma 1.1. (i) The map τ : G −→ W is a surjection with τ (1) = ², τ (g1 g2 ) = τ (g1 )g2
and τ (πg) = d(π)τ (g) for every g, g1 , g2 ∈ G and π ∈ P ; (ii) T² = Ker(d); (iii) j(g, z) =
1 for every g ∈ T² and z ∈ H; (iv) The mapping τ : G −→ W induces a bijection
ιτ : T² \G −→ W defined by T² g 7−→ τ (g) for g ∈ G.
2.A FORMAL THEORY OF EISENSTEIN SERIES. In this paragraph all of the
computations are formal and we ignore the convergence questions. Let Γ be a subgroup of
G and B be a complete set of representatives of the double-coset decomposition
P \G/Γ of
F
G with respect to (P, Γ). That is, there is the disjoint union G = β∈B P βΓ. For β ∈ B,
P ∩ βΓβ −1 becomes a transformation group on the set βΓ, where the action of P ∩ βΓβ −1
on βΓ is defined via left multiplication. Let Sβ be a complete set of representatives of
the orbits in βΓ relative to P ∩ βΓβ −1 . The proof of the following Lemma is elementary,
so it is omitted.
F
Lemma 2.1. (i) G = β∈B P Sβ ; (ii) The set W is the disjoint union of the sets M ²Sβ
F
and the map defined by (m, x) 7→ m²x is a bijection from β∈B M × Sβ to W .
Let ∆ be a subgroup of M (then ∆ is a transformation group on W , where the action
of ∆ on W is induced from that of M on W ). The orbits ∆\W in W relative to ∆ have
the following description:
F
Lemma 2.2. ∆\W = β∈B {r²x | r ∈ ∆\M, x ∈ Sβ }.
Assume that
(4)
d(P ∩ βΓβ −1 ) ⊂ ∆ ⊂ Ker(δ),
for every β ∈ B. So there is the canonical induced mapping δ∗ : ∆\M −→ G. As a
notation (which are clearly well-defined) introduce:
(5)
j(g, z) = j(τ (g), z) and j(x, z) = j(w, z),
for g ∈ G, x = ∆w ∈ ∆\W and z ∈ H. Take λ : W −→ C any map satisfying the
condition
(6)
λ(∆wγ) = ψ(γ)λ(w),
for any γ ∈ Γ and w ∈ W with a character ψ : Γ −→ T = {z ∈ C :| z |= 1} such that
(7)
P ∩ βΓβ −1 ⊂ Ker(ψ),
3
for every β ∈ B. We now introduce the following series:
(8)
E(., λ, ψ) : H −→ G defined by E(z, λ, ψ) =
X
λ(x)j(x, z),
x∈∆\W
for z ∈ H, which is a well-defined function by virtue of the assumption (6).
(9)
X
E (., Γ, ψ) : H −→ G defined by E (z, Γ, ψ) =
ψ(x)j(x, z),
x∈(P ∩Γ)\Γ
for z ∈ H, which is a well-defined function, since d(P ∩ Γ) ⊂ ∆ ⊂ Ker(δ) and P ∩ Γ ⊂
Ker(ψ). The series introduced in (9) is the Eisenstein series of G (with respect to Γ and
ψ : Γ −→ T). For β ∈ B, denote by ψβ : βΓβ −1 −→ T to be the character which factors
through
β −1 −conjugation
ψ
→ T.
ψβ : βΓβ −1 −−−−−−−−−−−→ Γ −
(10)
Consider the Eisenstein series E (z, βΓβ −1 , ψβ ), which is well-defined since d(P ∩βΓβ −1 ) ⊂
∆ ⊂ Ker(δ) and P ∩βΓβ −1 ⊂ Ker(ψ). Then for β ∈ B, the β-transform of E (., βΓβ −1 , ψβ ) :
H −→ G is given by
E ||β (z, βΓβ −1 , ψβ ) =
(11)
X
ψ(β −1 x)j(x, z), for z ∈ H.
x∈Sβ
We now state the main theorem: the series E(z, λ, ψ) is a linear combination of βtransforms of Eisenstein series E (z, βΓβ −1 , ψ) for β ∈ B. More precisely,
P
Theorem 2.3. For any z ∈ H, E(z, λ, ψ) = β∈B Lβ E ||β (z, βΓβ −1 , ψβ ) where Lβ =
P
r∈∆\M λ(r²β)δ∗ (r) for β ∈ B.
3. APPLICATION OF THE FORMAL THEORY OF EISENSTEIN SERIES. Let F be a totally real algebraic number field of degree m and K a CM -field
with the maximal totally real subfield F (i.e K is a totally imaginary quadratic extension
of F ). To make our exposition uniform, the symbol k will denote either F or K if there
is no fear of confusion and Ok will denote the ring of integers of k. We denote by a either
a F the set of archimedean primes of F or a fixed CM -type of K and h will denote the
∼
set of non-archimedean primes of F . We call ι the Galois involution of K/F . Let Gk be
the algebraic group defined over F by
∼
Gk = {X ∈ GL2n (k) | XJn X ∗ = Jn },
(12)
µ
¶
0 −1n
where Jn =
∈ M2n (Q) and X ∗ denotes respectively, the transpose t X in
1n
0
the totally real case k = F , and the transpose-conjugate t X ι in the CM -case k = K.
∼
We denote the group Gk ∩ SL2n (k) simply by Gk . Let P be the parabolic subgroup of
G corresponding to the partition (n, n), Q the Levi component and R the unipotent
∼
∼
∼
∼
radical of P . The subgroups P , Q and R of G are defined likewise. Let Wk = {α ∈
∼
Mn×2n (k) | αJn α∗ = 0, rank(α) = n}. The surjection τ : Gk −→ Wk defined by
4
τ (X) = (0n , 1n )X = (cX , dX ) for X =
∼
µ
aX
cX
bX
dX
¶
∼
∈ Gk induces a right transitive action
∼
∼
of Gk on Wk as τ (X)Y = τ (XY ) for X and Y elements of Gk . µ
Let d : P k −→
¶ GLn (k) be
∼
a X bX
the surjective homomorphism defined as d(X) = dX for X =
∈ P k . Then
0 dX
we have the commutative square (2). The group GLn (k) acts faithfully on Wk from the
∼
left by m(c, d) = (mc, md) for m ∈ GLn (k) and (c, d) ∈ Wk . The right action of Gk and
the left action of GLn (k) on Wk are clearly compatible. The isotropy subgroup T(0n ,1n )
∼
∼
∼
∼
of (0n , 1n ) in Gk is the unipotent radical Rk of P k . The local archimedean group Gk,ν
(ν ∈ a) acts on the symmetric space
½
{Z ∈ Mn (C) | t Z = Z, Im(Z) > 0}, k = F ,
Hk =
k = K,
{Z ∈ Mn (C) | i(t Z − Z) > 0},
∼
∼
as α(Z) = (aα Z + bα )(cα Z + dα )−1 for α ∈ Gk,ν and Z ∈ Hk . The map µν : Gk,ν × Hk −→
∼
GLn (C) defined by µν (α, z) = cα z + dα is a GLn (C)-valued factor of automorphy of Gk,ν
on Hk . Following the notation of [6], we introduce
Y
(13)
Jk,s (x, Z) =
det(µν (xν , zν ))k | det(µν (xν , zν )) |s ,
a
ν∈a
where Z = (zν )ν∈aa ∈
(14)
Hak ,
x ∈ G ,→ GA , k ∈ Z and s ∈ C. It is clear that
∼
Jk,s (π, Z) = Nk/Q (det(d(π)))k/[k:F ] | Nk/Q (det(d(π))) |s/[k:F ] , for π ∈ P k .
So the morphism δ : GLn (k) −→ C× , δ(g) = Nk/Q (det(g))k/[k:F ] | Nk/Q (det(g)) |s/[k:F ] for
∼
g ∈ GLn (k) satisfies δ(d(π)) = Jk,s (π, Z) for every π ∈ P k and Z ∈ Hak . Hence we have
∼
∼
shown that: {Gk , P k , GLn (k), Wk , (0n , 1n )} is an Eisenstein datum.
∼
For a congruence subgroup Γ of Gk (cf [6] pp 424), define the Eisenstein series
X
(15)
E (Z, s; k, Γ) =
Jk,s (α, Z)−1
α∈(Pk ∩Γ)\Γ
Hak ,
Q
s ∈ C, k ∈ Z; provided that ν∈aa det(µν (xν , zν ))k = 1 for
for Z = (zν )ν∈aa ∈
every x ∈ Pk½∩ Γ to make the sum meaningful. The series E (Z, s; k, Γ) converges for
n + 1 (k = F )
Re(k + s) >
. Let ∆ be a sufficiently small congruence subgroup of
2n
(k = K)
GLn (k) such that δ(∆) = 1. Let S(V ) denote the space of locally constant functions
V −→ C on a vector space V over Q. Recall that a function ` : V → C is called
locally constant, if there exist two lattices L1 and L2 of V so that `(x) = 0 if x ∈
/ L1
and `(x) = `(y) if x ≡ y (mod L2 ). Consider an element λ ∈ S(Mn×2n (k)), such that
λ : Mn×2n (k) −→ C satisfies the condition λ(∆wΓ) = λ(w) for w ∈ Mn×2n (k). Following
the notation (5), introduce the sum
X
(16)
E(Z, λ) =
λ(x)Jk,s (x, Z)−1 ,
x∈∆\Wk
for Z ∈
Hak .
Theorem 2.3 yields E(Z, λ) =
∼
∼
P
β∈B
Lβ E ||β (Z, s; k, βΓβ −1 ), where B
is a complete
set of representatives of P k \Gk /Γ which is known to be a finite set and
P
Lβ = r∈∆\GLn (k) λ(rτ (β))Nk/Q (det(r))−k/[k:F ] | Nk/Q (det(r)) |−s/[k:F ] . Such Dirichlet
series are studied by Shimura in [7] (pp. 309-313). We state the main theorem of this
paragraph. The proof utilizes Proposition 9.2 of [7].
5
∼
Q
Theorem 3.1. Let Γ be a congruence subgroup of Gk satisfying the condition ν∈aa det(µν (xν , zν ))k =
1 for every x ∈ Pk ∩ Γ and Z = (zν )ν∈aa ∈ Hak and ∆ be a sufficiently small congruence
subgroup of GLn (k) such that δ(∆) = 1. Let λ : Mn×2n (k) −→ C be a locally constant
function with the property λ(∆wΓ) = λ(w) for w ∈ Mn×2n (k). Then, the series E(Z, λ)
is a finite linear combination of β-transforms of the Eisenstein series E (Z, s; k, βΓβ −1 )
whose coefficients are sums of products of certain Hecke L-series. More precisely,
E(Z, λ) =
X
β∈B
2
4
X
1≤j≤k(β)
s
[k:F ]
aβj bβj
3
s
− i, χn−i ψβji )5 E ||β (Z, s; k, βΓβ −1 )
L(
[k
:
F
]
0≤i≤n−1
Y
where aβj ∈ C, 0 < bβj ∈ Q and ψβji are Hecke characters of k×
A of finite order for β ∈ B,
1 ≤ j ≤ k(β) and 0 ≤ i ≤ n − 1.
If Re(s) is sufficiently large, then all the series introduced in this paragraph are absolutely convergent and the computations make sense. The proof of the fact that Eisenstein
series E (Z, s; k, Γ) can be recovered from the series of the type E(Z, λ) is much more involved and will be published as a separate paper.
References
1. A. Borel, Introduction to automorphic forms, Algebraic Groups & Discontinuous Subgroups. Proc.
Sympos. Pure Math of the A.M.S (A. Borel & G. Mostow eds.) IX (1966), 199-210.
2. P. Feit, Poles and residues of Eisenstein series for symplectic and unitary groups, Memoirs of the
A.M.S 346 (1986).
3. K. I. Ikeda, Linear equivalence of Eisenstein series, thesis, Princeton University (1993).
4. R. P. Langlands, Euler products, Yale University, Mimeographed Notes (1970).
5.
, On the functional equations satisfied by Eisenstein series, Lecture Notes in Mathematics,
Springer Verlag 544 (1976).
6. G. Shimura, On Eisenstein series, Duke Math J. 50 (1983), 417-476.
7.
, Differential operators and the singular values of Eisenstein series, Duke Math J. 51 (1984),
261-329.
Division of Arithmetic, Department of Mathematics, Research Institute for Basic Sciences, TÜBİTAK-Marmara Research Center, P.O Box 21, 41470 Gebze, Turkey
E-mail address: [email protected] Fax: +90 262 641 2309 Phone:+90 262 641 2300
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